Elasticity of Ferropericlase across the Spin Crossover in the Earth’s Lower Mantle

Knowing the elasticity of ferropericlase across the spin transition can help explain seismic and mineralogical models of the lower-mantle including the origin of seismic heterogeneities in the middle to lowermost parts of the lower mantle1234. However, the effects of spin transition on full elastic constants of ferropericlase remain experimentally controversial due to technical challenges in directly measuring sound velocities under lower-mantle conditions12345. Here we have reliably measured both VP and VS of a single-crystal ferropericlase ((Mg0.92,Fe0.08)O) using complementary Brillouin Light Scattering and Impulsive Stimulated Light Scattering coupled with a diamond anvil cell up to 96 GPa. The derived elastic constants show drastically softened C11 and C12 within the spin transition at 40–60 GPa while C44 is not affected. The spin transition is associated with a significant reduction of the aggregate VP/VS via the aggregate VP softening because VS softening does not visibly occur within the transition. Based on thermoelastic modelling along an expected geotherm, the spin crossover in ferropericlase can contribute to 2% reduction in VP/VS in a pyrolite mineralogical model in mid lower-mantle. Our results imply that the middle to lowermost parts of the lower-mantle would exhibit enhanced seismic heterogeneities due to the occurrence of the mixed-spin and low-spin ferropericlase.


Modelling the Fraction of the High-Spin and Low-Spin States in Ferropericlase
Following the modelling procedures reported in previous studies 1- 3 , we have used the pressure-volume 4 relations from our X-ray diffraction measurements to evaluate the fraction of the high-spin (HS) and low-spin (LS) states of iron in ferropericlase (Mg0.92Fe0.08)O as a function of pressure and temperature (P-T) (Figs. S1 and S2). The P-V data of the ferropericlase was initially compared with the P-V relation of the end-member MgO as a starting reference at high pressures and room temperature 4 . The comparison permits us to clearly evaluate the volume reduction over the pressure range across the spin transition 2 . Since the HS ferropericlase exhibits a similar equation of state (EoS) behavior to that of MgO, such comparison also helps establish the EoS parameters for the HS state 2 . With the width of the transition and the thermal EoS parameters of the HS state initially established, the fraction of the LS state (nLS) at a given P-T condition can be obtained using the following equations 1, 3  ∆ ( , ) * = 0 ( ) + 1 ( ) [2] = − − [3] where ∆G(P,T) ⃰ is the difference in Gibbs free energy between the LS and HS states, Pn is the normalized pressure as determined by the ending pressure of the HS state (PHS) and the onset pressure of the LS state (PLS), and b0 and b1 are two temperature-dependent constants. Using the non-linear least squares fit of the nLS to the P-V data at 300 K, we have obtained b0 = 1220 (25) and b1 = −2341 (46).
Based on the solid-solution mixing of the HS and LS states in the ferropericlase lattice as well as the derived LS fraction (nLS), the unit cell volume of ferropericlase (V) across the spin transition is expressed as the ratio between the unit cell volume of the HS state (VHS) and the LS state (VLS) at a given pressure and 300 K (Fig. S2) 3 : It follows that the isothermal bulk modulus (KT) of the system across the spin transition can be described using the ratio of the HS and LS states 3 : where KHS and KLS are the KT of the HS and LS state, respectively, ρ is the density, and VΦ is the bulk sound velocity (Figs. S2 and S3).

Derivation of the Full Elastic Constants of the Single-Crystal Ferropericlase
The ferropericlase platelet with (100) orientation allows us to measure VP velocities as well as VS velocities with <110> polarization along principle [100] and [110] crystallographic axes using the ISS and BLS techniques, respectively, in a DAC. Together with the P-V data from synchrotron X-ray diffraction measurements (Figs. S1 and S2), here we have combined the VS data from BLS measurements and the VP data from ISS measurements ( Fig. 1) to derive the elastic constants (C11, C12, C44) of the single-crystal ferropericlase using the following equations 5 ( Fig. 2 and Fig. S6): [100] = ( 11 / ) 1/2 [7] [100] < 110 >= ( 44 / ) 1/2 [8] [110] = [( 11 + 12 + 2 44 )/2 ] 1/2 [9] [110] < 110 >= [( 11 − 12 )/2 ] 1/2 [10] where [uvw] represents the crystallographic direction for the acoustic wave propagation, and <uvw> indicates the polarization direction. Since the method for deriving the full elastic constants involves multiple experimental data sets and the use of multiple equations listed above, the elastic constants reported here are derived from internally-consistent numerical iterations through minimization of the uncertainties in the derived parameters using the aforementioned equations as well as the finite-strain equations discussed below 6  = − 1 2 | , [16] where G is the total Gibbs free energy of the system, and σi and σj are the ith and jth stress components, respectively, in the Voigt notation. The elastic compliances for the cubic ferropericlase are given as 9 : The relationships between the elastic constants and the compliances are described as 10  Comparison of the modelled elastic constants with experimental results shows a good agreement with each other in the HS, MS, and LS states, validating the thermoelastic theory for the elasticity of ferropericlase across the spin transition 9 (Fig. 2).

Comparison of the Velocity Results at High Pressures
Here we compare our results with the ones reported in previous studies using similar BLS  11 , but are slightly higher within the spin transition and in the low-spin state (Fig. S5). The difference within the spin transition can be explained as a result of different iron contents in these measurements as higher iron content is expected to contribute to a stronger effect on the velocity.
Comparison of our measured VP with previous BLS measurements below 20 GPa shows great consistency within uncertainties 12 , whereas there is a significant discrepancy within the spin transition and in the low-spin state (Fig. S5). Our present results show a stronger VP softening within the spin transition and a lower VP in the low spin state than that in the previous ISS measurements for ferropericlase with 6% iron. These differences may be explained by the different iron contents, pressure media, as well as experimental uncertainties including the orientation of the crystal used in the experiments.

Modelling Thermoelastic Parameters across the Spin Crossover in the Lower Mantle
Using the experimentally-derived thermal EoS parameters and the elastic constants at high       study; black squares: IXS study with 17% iron content 19 ; green circles: BLS study below 20 GPa with 6% iron content 12 ; blue triangles: ISS measurements with 6% iron 18 ; dark cyan down triangles: BLS study with 10% iron 11 ; magenta stars: ultrasonic measurements with 8% iron 20 ; blue and green lines: theoretical results with 18.75% and 10% iron content, respectively 9 .